Nonliner Properties Of The One-dimensional Oscillator With Quantized Impedance Based On Special Relativity

With the development of pulsed laser technology,the pulse width is compressed and the peak power is continuously increased.Meanwhile the frequency of the laser moves towards the frequency of the soft X ray.When ultra-fast lasers interact with the medium,the average velocity of electrons is likely to approach the speed of light.Under such conditions,relativistic effects need to be considered.Generally speaking,the theoretical method of studying relativistic effect by means of quantum theory should be to solve Dirac equation.However,in addition to a few cases,it is difficult to obtain the analytical solution of the equation.Based on the special relativity dynamics equation,the nonlinear properties of one-dimensional oscillator with quantized impedance are studied in this paper by taking hydrogen atom as an example.The emphases are on the influences of relativistic correction of the electron mass on the third harmonic and two-photon absorption properties.Comparing with the method solving the time-dependent Dirac equation directly,this method is relatively easy.And the physical model of this method is manifest at the same time.The main achievements of this paper are as follows.1.The approximate analytical solution of third order susceptibility which is related to the third harmonic and two photon absorption of one-dimensional oscillator with quantized impedance is derived theoretically in the frequency domain by way of Fourier transform and Taylor expansion.The results show that the relativistic effect of the oscillator motion will affect both the two kinds of the third order susceptibility.And the variation is in the proportional relationship to the fourth power of the laser frequency.2.The influences of relativistic correction of the electron mass on the third harmonic intensity are quantitatively analyzed.The third order susceptibility squared curves corresponding to the third order harmonic in the vicinity of the linear resonance of the hydrogen atom Lyman system are numerically simulated under the relativistic condition.The peak values show that when the laser frequency is 1.84×10¹⁶rad/s,the enhancement rate of third harmonic intensity at the peak values caused by the variation of electronic relativistic mass is 6.1%under the transition from n=1 to n=3?n_?=1?.And when the laser frequency is 1.94×10¹⁶rad/s,the enhancement rate is up to 391.8%under the transition from n=1to n=4?n_?=1?.That means with the increasing of laser frequency,the relativistic mass correction shows an obvious reinforce effect on the third harmonic intensity.3.The influences of relativistic correction of the electron mass on TPA are quantitatively analyzed.The imaginary part of the third order electric susceptibility curves corresponding to TPA in two cases of relativity and non-relativity are simulated with taking Lyman series of hydrogen atom as an example.The results indicated that relativistic mass correction has little effect on TPA of Lyman series of hydrogen atom,so it can be usually neglected.